Capacity of the α-Brjuno-Rüssmann set
Nurali Akramov
TL;DR
The paper investigates the thinness of Brjuno-type exceptional sets for small denominators via potential-theoretic capacity. It links the α-Brjuno-Rüssmann sets $BR_α$ to the Diophantine sets $\mathcal{A}(β,γ)$ through continued-fraction data and demonstrates zero $C_\sigma$-capacity for the complements by constructing a finite measure on rationals and analyzing potentials with kernels $k^1_\sigma$ and $k^2_\sigma$. The main results show that the complements of the Brjuno set and the Perez-Marco set have zero capacity with explicit kernels when $σ>2$, and derive a corollary for the α-Brjuno–Rüssmann case, with corresponding $h$-Hausdorff-measure conclusions. These findings quantify the scarcity of resonant parameters in quasi-periodic Hamiltonian problems and connect complex-analytic capacity to arithmetic properties of irrational numbers.
Abstract
In this work, we prove that the complement of the Brjuno set $\mathcal{B}$ has a zero capacity with respect to the kernel $k^1_σ(z,ξ)=\ln^2{|z-ξ|}\left|\ln{\ln{\left(e+\frac{1}{|z-ξ|}\right)}}\right|^σ$ for any$σ> 2$. Similarly, the complement of the Perez-Marco set $\mathcal{PM}$ has a zero capacity with respect to the kernel $k^2_σ(z,ξ) = \ln^{2}{\ln\left(e+\frac{1} {\left| {z - ξ}\right|}\right)}\cdot\ln^σ{\ln\ln\left(e^3+\frac{1} {\left| {z - ξ}\right|}\right)}$ for any $σ>2$.